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 object 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...

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...

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 object 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 very welcome too. I am getting fricasseeing this morning with 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...

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 object 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...