Suppose $\psi: \mathbf{Groups} \rightarrow \mathbf{Sets}$ is a left adjoint functor. How would I go about evaluating $\psi(\mathbb{Z})$?
Since $\psi$ is left adjoint, let $\psi$ be left adjoint to a functor $\varphi: \mathbf{Sets} \rightarrow \mathbf{Groups}$. Then by definition we have a bijection between $\textrm{Mor}_{\mathbf{Groups}}(\mathbb{Z}, \varphi(S)) \overset{\sim}{\rightarrow} \textrm{Mor}_{\mathbf{Sets}}(\psi(\mathbb{Z}), S)$ for all sets $S$. Maybe it would help to study $\textrm{Mor}_{\mathbf{Groups}}(\mathbb{Z}, \varphi(S))$ but I am unsure how to proceed.
More generally, let $C$ be a category with a zero object $0$ and let $D$ be a category with a strictly initial object $\emptyset$. Then any left adjoint functor $F : C \to D$ maps everything to (something isomorphic to) $\emptyset$. In fact, if $x \in C$, we have a morphism $x \to 0$. Since left adjoints preserve initial objects, the image is a morphism $F(x) \to \emptyset$, which has to be an isomorphism by the choice of $\emptyset$.