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](http://mathoverflow.net/questions/32791/how-is-the-right-adjoint-f-to-the-inverse-image-functor-f-described-for-f/32808#32808).

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.