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 $G$. (Once you have the initial morphisms, this extension is really unique, i.e. not just unique up to natural isomorphism.)
Note that this theorem does not involve having an "initial guess" for the left adjoint, but actually constructs it for you from the limited data of the initial morphisms. It is not hard to prove, but not trivial, and is explained relatively well here.
Once you know this, you can really take confidence in "follow your nose"-style adjoint construction. It doesn't even involve having an initial guess for the left adjoint (as a functor), since the theorem constructs it for you from the l 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.