Would it be possible to enlighten me (or even better give a reference) about enough projectives (injectives) in functor categories?


Here is a precise question. Let $C$ be a small category, whose total morphism set has cardinality $\alpha$. Let $A$ be an abelian category with enough projectives (dually, injectives) and coproducts (products) up to cardinality $\alpha$. The functor category $A^C$ is clearly abelian.


**Question** Does $A^C$ have enough projectives (dually, injectives)?


A reference would be ideal but an explanation would be very welcome too. I am getting fricasseeed this morning by this question: surely, the issue should be whether one can cover an object in $A$ by a projective object functorially but I have no idea how to spell it out...



Projectives and Injectives in Functor Categories
For each object $c$ in $\mathcal{C}$, let $c^\* : [\mathcal{C}, \mathcal{A}] \to \mathcal{A}$ be evaluation at $c$. It is an exact functor, so if a left adjoint $c\_! : \mathcal{A} \to [\mathcal{C}, \mathcal{A}]$ exists, $c\_!$ will preserve projective objects. Assume $\mathcal{C}$ has $\le \alpha$ morphisms and $\mathcal{A}$ has coproducts for families of $\le \alpha$ objects. Then the left adjoint $c\_!$ exists and can be computed the following formula:
$$(c\_! A) (c') = \mathcal{C} (c, c') \odot A$$
(Here, $X \odot A$ denotes the coproduct of $X$-many copies of $A$.)


Now, let $F$ be an object in $[\mathcal{C}, \mathcal{A}]$. For each object $c$ in $\mathcal{C}$, choose a projective cover $P\_c \twoheadrightarrow F c$. By adjointness, we obtain morphisms $c\_! P\_c \to F$ in $\mathcal{A}$; note that the composite $P\_c \to c^\* c\_! P\_c \to F c$ is the epimorphism we started with, so $c^\* c\_! P\_c \to F$ is an epimorphism in particular. Now, form the object $P = \bigoplus\_{c \in \operatorname{ob} \mathcal{C}} c\_! P\_c$; then there is a morphism $P \to F$ such that the components $P c \to F c$ are all epimorphisms. Furthermore, each $c\_! P\_c$ is projective, and the class of projective objects is closed under coproducts, so we have obtained the required projective cover of $F$.


I'm afraid I do not have a reference, but the above is essentially the same as the proof that (say) $[\mathcal{C}, \mathbf{Ab}]$ has enough projectives. It can be further generalised to the case where $\mathcal{C}$ is preadditive and $[\mathcal{C}, \mathcal{A}]$ is the category of additive functors.


