14 added 41 characters in body

Yes, and it is easy

I think frequently the easiest to check . The sufficient conditions are the followingresult makes , which also make precise why "this looks like it might work" is so often successful:

Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(I_Y)$ from $Y$ to $G$. Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto I_Y$ extends in a unique way to act on morphisms defining a functor $F: D\to C$, which moreover is left adjoint to the original functor $G$.

This is well-known and easy to prove (well, depending on who you ask), but is non-trivial and involves many steps, which are explained relatively well here. (Essentially one is recovering the entire adjoint situation from just one functor and a unit transformation.)

Once you know it, you can really take confidence in "follow your nose"-style adjoint construction. It doesn't involve having an "initial guess" for the left adjoint (as a functor), but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms --- really unique, not just up to natural isomorphism.

As an example of how this can be useful, think of the inclusion functor $U$ from $AbGrp$ to $Grp$. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H]$ in $AbGrp$ with a map
$H\to Ab(H)$ satisfying an initial (universal) property. But then by the above theorem, we can automatically extend this association in a unique way to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.

This same trick expedites the construction of adjoints in pretty much any situation you can think of.

Edit: Sometimes this theorem is used as an alternative definition for adjoint functors in terms of universal morphisms. However you look at it, the real utility is knowing that this "weak", and in fact asymetric, condition actually implies the "stronger", symmetric definitions of adjoints via hom-sets or units/counits. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.

13 compactified

Yes, and it is easy to check! In fact, the . The following theorem result makes precise why "this looks like it might work" is so often successful:

This result is well-known , and easy to prove (well, depending on who you ask), but is non-trivial and involves many steps, which are explained relatively well here. (Essentially one is recovering the entire adjoint situation from just one functor and a unit transformation.)

Once you know it, you can really take confidence in "follow your nose"-style adjoint construction. It doesn't involve having an "initial guess" for the left adjoint (as a functor), but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms --- really unique, not just up to natural isomorphism.

This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.

Notice that:

(1) This theorem does not involve having an "initial guess" for the left adjoint (as a functor), but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms (I think it's really unique, not just up worthwhile to natural isomorphism).

(2) Its conditions are asymmetric, but the conclusion and sift through the definition three different characterizations of adjoints (using hom-sets) is symmetric, which means there really is something to prove here.

(3) It follows further from this theorem that the functor $G$ actually defines provides you with a family of terminal morphisms to the functor $F$, which is now many logical steps removed from the assumption that initial morphisms existed from $G$.

(4) The initial morphisms in the theorem end up forming the unit of the adjunction, but we do not start off assuming these morphisms are natural tranformations. Instead, the functor $F$ is constructed in a way to make them natural, and the counit (ie terminal morphisms) end up existing for free, and satisfy the required equationsgiven on Wikipedia.

12 added 1 characters in body

Yes, and it is easy to check! In fact, the following theorem makes precise why "this looks like it might work" is so often successful:

Theorem: A functor $G: C\to D$ is a right adjoint functor (i.e. has a left adjoint) if and only if for each object $Y$ in $D$, there exists an initial morphism $\phi_Y:Y\to G(I_Y)$ from $Y$ to $G$. Moreover, once you find such an initial morphism from each $Y$ to $G$, the association $Y\mapsto I_Y$ extends in a unique way to act on morphisms defining a functor $F: D\to C$, which moreover is left adjoint to the original functor $G$.

This result is well-known, easy to prove, but is non-trivial and involves many steps, which are explained relatively well here. (Essentially one is recovering the entire adjoint situation from just one functor and a unit transformation.) Once you know it, you can really take confidence in "follow your nose"-style adjoint construction.

As an example of how this can be useful, think of the inclusion functor $U$ from $AbGrp$ to $Grp$. It's easy to see that any group $H$ has an abelianization $Ab(H) = H/[H,H]$ in $AbGrp$ with a map
$H\to Ab(H)$ satisfying an initial (universal) property. But then by the above theorem, we can automatically extend this association in a unique way to act on morphisms as well, defining an abelianization functor $Ab$ which is left adjoint to the inclusion $U$.

This same trick expedites the construction of adjoints in pretty much any situation you can think of. I think it's really worthwhile to sift through the three different characterizations of adjoints given on Wikipedia.

Edit: Sometimes this theorem is used as an alternative definition for adjoint functors in terms of universal morphisms. However you look at it, the real utility is knowing that this "weak", and in fact asymetric, condition actually implies the "stronger", symmetric definitions of adjoints via hom-sets or units/counits.

Notice that:

(1) This theorem does not involve having an "initial guess" for the left adjoint (as a functor), but actually constructs it for you in a way that is uniquely determined by the limited data of the initial morphisms (really unique, not just up to natural isomorphism).

(2) Its conditions are asymmetric, but the conclusion and the definition of adjoints (using hom-sets) is symmetric, which means there really is something to prove here.

(3) It follows further from this theorem that the functor $G$ actually defines provides you with a family of terminal morphisms to the functor $F$, which is now many logical steps removed from the assumption that initial morphisms existed from $G$.

(4) The initial morphisms in the theorem end up forming the unit of the adjunction, but we do not start off assuming these morphisms are natural tranformations. Instead, the functor $F$ is constructed in a way to make them natural, and the counit (ie terminal morphisms) end up existing for free, and satisfy the required equations.