"Quick, attentive, reserved with strangers, but he can be gentle and affectionate with those he's willing to accept. Fault - Excessively timid, hysterical, or aggressive character"
Thus reads the temperament section of the AKC Azawakh standard. The phrasing is similar in Azawakh standards all around the world. It's brief and those adjectives could have a huge variety of meanings, so how should they be applied to the Azawakh in front of you?
Quick - Azawakh are quick to judge, quick to react, quick to think for themselves. In running they are quick to turn directions and quick off the line for a sighthound designed for endurance. When an Azawakh does decide to leap into action, it will be done with haste. With any dog that is quick to take action, we should thus accordingly comport ourselves in a calm, smooth, unhurried fashion. Treat the dog in the manner you wish them to respond, and it will help them respond in turn.
Attentive -
Any Azawakh owner that has their dog out in public has likely heard from passersby "They sure keep their eye on everything, don't they?" I once had a rally obedience judge that also was approved to judge Azawakh in conformation laugh after our run. "He sure kept an eye on me the whole time, as he should."
An Azawakh's head is on a swivel, taking in all the stimuli around them. Their ears might perk, their body might orient toward a handler running , a bit late for their ring time. A judge crossing their arms and scrutinizing them closely may look a threat. At the sound and smell of a dog coming up hard behind them, they might jump and turn to face the stranger. They are attentive to their environment, but what's more, they are attentive to their handler. Is the handler relaxed, do they accept the strange dog or person? An Azawakh that attends to the people they trust is a correct Azawakh.
Reserved with strangers -
Merriam-Webster defines reserved as "restrained of words and actions" as well as "kept or set apart". An Azawakh is certainly restrained of actions while they attend the environment and then decisively take action. If you meet an Azawakh and feel as though you are being weighed and judged, it's certainly because you are. A reserved character is decidedly incompatible with attention-seeking behavior. They are a breed that holds themselves apart from the crush of activity around them and when that crush descends upon them with no regard for their desire to understand their surroundings before interacting with them, their tendency will be to take action immediately.
Gentle and affectionate with those they are willing to accept -
The standard puts the onus of acceptance on the Azawakh. The standard grants the Azawakh a choice in choosing to not accept a stranger. Within the context of a western society, that acceptance is not necessarily (and likely is not) granted to all strangers that wish to touch them. Within the context of a show ring, an Azawakh should accept a judge handling them because they trust and accept their handler, not inherently because the Azawakh truly accepts the judge.
When an Azawakh accepts you, when you become one of their people, there is nothing like it. You fall into their soulful eyes, they soften, they lean into you. If they truly trust a person, an Azawakh would walk through fire or lay down their life for them.
Fault - Excessively timid, hysterical, or aggressive character -
It should be notable that this section is only listed as a fault rather than a serious fault or even disqualification. I will never propone an unstable, unpredictable Azawakh. They shouldn't be in public spaces and they certainly should not be in the show ring. However, the standard acknowledges that the temperament described within it is not one of a gregarious, stranger social dog, but rather of a dog that thinks, that assesses. It is a description of a dog that, when rushed, can be quick to take action.
Especially with young dogs who are new to the show world and slow to warm up to the pressure of a person descending on them formally, there may be a learning curve. They may resist the exam before settling. With puppies and adolescents of all breeds, granting grace is a favorable practice.
However, regardless of age, if the dog is unpredictable, aggressive, unsafe, and unthouchable, the dog must be excused or dismissed in accords with the severity of the offense.
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Absent from the history section of the standard, is the background and original (and continued) purpose of Azawakh. In their countries of origin, Niger, Mali, and Burkina Faso, they are first and foremost pack dogs that alert and defend their people, livestock, and camp -- they are primarily guardians. Game is scarce in the region, so while they are coursing hounds, that is, and has long been, a secondary focus for Azawakh. Their origin informs us how this interesting and rather uncommon temperament came to be.
I once read a fascinating article by a dog trainer and Labrador breeder explaining the difference in temperament between a Labrador Retriever and a Golden Retriever. The author explained that while Goldens are retrieving dogs, they are unlike the Labrador Retrievers who are expected to break trail and push through ice when retrieving out of water. This explanation instantly explained so many pieces of why breeds that must break trail and push through environmental adversity are the first to push through baby gates and doors to get to their humans or food or any tempting stimulus they may desire. They push through environmental obstacles, because that is what they have been bred to do over the course of many generations.
Thus can Azawakh be understood, by understanding the pressures of their indigenous environment. Unlike a dog that has the benefit of bulk and size behind them, Azawakh are well aware that they are thin, consummately adapted to their harsh, hot environment. They rely on pack and tribe to back them in case of a threat. While a western livestock guardian may stand their ground upon first sign of a novel presence, an Azawakh will generally move away to observe. They will yield to pressure because they are aware of their size, but they will not flee, as most sighthounds would. They should assess the stranger and if that stranger proves to be a threat, then they will stand their ground, trusting their pack and people to back them.
An Azawakh that lashed out at people simply walking by would not last long in their village. Similarly, an Azawakh that greeted the novel with enthusiasm would counter their original purpose and likely not survive to reproduce. Stability is key. They should discern the harmless from a threat. Sensibly suspicious certainly, but predictable and manageable in their reactions.
The temperament of the breed has been forged by their harsh, hot environment and the sands of time spanning millennia in exactly the same way they have developed their rangy, dry, lean appearance that makes them uniquely suited to thermoregulation.
Even should breeders, for some reason, wish to change their temperament to something more friendly and welcoming, they will be fighting millennia of ingrained instincts created and honed for very specific reasons. I once heard a Sussex Spaniel breeder, one of the oldest spaniel breeds in existence, speak about taking her dogs hunting after generations of not being in the field. She found that their hunting instincts, so ingrained over the course of a couple hundred years, were still very much intact in spite of disuse. Imagine how much more the instincts of a breed thousands of years older must be then.
Until the 1970s, only fifty years ago, every single Azawakh in existence was a working dog, because Azawakh had not yet been imported out of the Sahel. They protected their people and their camp and stock. They caught game for their people's dinner. They traveled alongside their nomads, vigilant and unwavering. This is a breed of those who wish a fierce and loyal companion. A breed for those who wish an ever constant companion that should neither seek attention from strangers nor lash out at those just going about their life. While I personally have found them very adaptable, why should we subject them to the stress of constant dinner parties and play dates, when there are people who wish their homes to be a refuge from the daily expectations of the outside world and when they venture out, they can venture out with a constant companion who cares nothing of outsiders.
Truly, Azawakh are a working dog in a sighthound suit and in the correct homes they make singularly devoted companions.