Whenever $A$ has all small (co)limits, small (co)limits in functor categories $A^C$ are pointwise: a cone $F : I \to A^C$ is a (co)limit iff each of its “components” or “partial evaluations” $F(c) : I \to A$ is a (co)limit.

But the property of being a pointwise (co)limit is obviously preserved by composition with $f:C' \to C$. So $f^*$ is continuous and co-continuous (and you can apply the AFT).

On the other hand, as Martin says in comments, we don't really need the AFT: the adjoints are given by left and right Kan extensions, which we can write down explicitly as (co)limits, as described by Michael Warren in this recent answer.

All of this is discussed in the chapters on limits and Kan extensions in Mac Lane Categories for the working Mathematician. Also: we don't need the local presentability for any of this, and the limits and colimits halves each work independently of the other.

Whenever $A$ has all small (co)limits, small (co)limits in functor categories $A^C$ are pointwise: a cone $F : I \to A^C$ is a (co)limit iff each of its “components” or “partial evaluations” $F(c) : I \to A$ is a (co)limit.

But the property of being a pointwise (co)limit is obviously preserved by composition with $f:C' \to C$. So $f^*$ is continuous and co-continuous (and you can apply the AFT).

On the other hand, as Martin says in comments, we don't really need the AFT: the adjoints are given by left and right Kan extensions, which we can write down explicitly as (co)limits, as described by Michael Warren in this recent answer.

All of this is discussed in the chapters on limits and Kan extensions in Mac Lane Categories for the working Mathematician. Also: we don't need the local presentability for any of this, and the limits and colimits halves each work independently of the other.