This question is inspired by Example 2.3.13 in Weibel's Homological Algebra book. Perhaps it is too elementary for MO, but hopefully someone can give me a quick answer.

The example is the following: let $I$ be a small category and $A$ an abelian category and $A$ is complete. Let $M \in A^I$ be a functor $M: I \to A$ (morally a module over $I$, not sure about the terminology). For any $k \in I$, we have the $k^{th}$ coordinate $ev_k: A^I \to A$ given by $M \mapsto M(k)$. The construction in the example gives a right adjoint to $ev_k$, denoted as $k_*: A \to A^I$, namely for any $a \in A$, we define $$ k_*(a)(i) = \prod_{\hom_I(i,k)} a $$ This seems to me to be like a skyscraper sheaf over the point $k$.

My question is the following: Let $I, J$ be small categories, and $A$ a complete abelian category. Fix a functor $F: J \to I$, then we have the pull-back functor $F^*: A^I \to A^J$. Can we define the right adjoint $F_*$ of $F^*$? That is, for any $M \in A^I$, $N' \in A^J$, we have $$Hom_{A^J}(F^*(M), N') = Hom_{A^I}(M, F_*(N'))$$