Csdrhrt

Let $A$ be a category and $C= Fun(A, Set)$ (i.e. the objects are functors and morphisms are natural transformations between them). I want to know if this category has a product. For given $X in A$ and functors $F_i in C$, is $Pi_{i in Lambda}F_i(X)$ the correct product? If yes, what would be the projection maps onto the $F_i$'s? I'm having a hard time constructing the required unique natural transformation. Any help is appreciated.

Yes, the product of $F_iintext{Obj}(text{Fun}(A,mathbf{Set}))$, $iinLambda$, is the functor $prod_{iinLambda}F_i$, such that $(prod_{iinLambda}F_i)(X)=prod_{iinLambda}(F_i(X))$ for every $Xintext{Obj}(A)$ and $(prod_{iinLambda}F_i)(f)=prod_{iinLambda}(F_i(f))$ for every $fintext{Mor}(A)$. Projections are natural transformations $P_jcolon(prod_{iinLambda}F_i)to F_j$, $jinLambda$, such that $P_j(X)=p_jcolon(prod_{iinLambda}(F_i(X)))to F_j(X)$, $Xin text{Obj}(A)$, $jinLambda$, where $p_j$ is a projection of an "ordinary" product in $mathbf{Set}$. These transformations are natural by the universal properties of products (note, that $p_jcirc(prod_{iinLambda}(F_i(f)))=F_j(f)circ p_j$, and the morphism $prod_{iinLambda}(F_i(f))$ is unique with such property). Such products are called pointwise by the obvious reason.

Oskar already gave a good answer (+1). I just wanted to add a different perspective.

Given a collection of objects $C_i$ in a category $C$, the product of the $C_i$, $prod_i C_i$, is an object representing the (contravariant) functor $Cto newcommandSet{mathbf{Set}}Set$
$$A mapsto prod_inewcommandHom{operatorname{Hom}} Hom(A,C_i).$$

Thus if $C$ and $A$ are categories, if $F_i$ are functors in $C^A$, the product of the $F_i$ is the functor $F$ representing
$$G mapsto prod_iHom_{C^A}(G,F_i).$$
I.e. it is a functor satisfying
$$Hom_{C^A}(G,F)simeq prod_i Hom_{C^A}(G,F_i).$$
However if we already know that $C$ has products, then let $F'(x)=prod_i F_i(x)$ for any $xin A$.

Then $$Hom_C(y,F'(x)) simeq prod_i Hom_C(y,F_i(x))$$ as functors from $C^{mathrm{op}}times Ato Set$. Thus for any functor $G:Ato C$, given a natural transformation $mu:Gto F'$, for any $xin A$, we can use this natural isomorphism to break apart the component $mu_x:G(x)to F'(x)$ of the natural transformation into maps $mu_{x,i} : G(x)to F_i(x)$ giving natural transformations $mu_i$, and conversely given natural transformations $mu_i : Gto F_i$, we can put them together to get a natural transformation $mu$. Thus $F'$ satisfies
$$Hom_{C^A}(G,F')simeq prod_iHom_{C^A}(G,F_i),$$
so $F'=prod_i F_i$ as desired.

It should hopefully be clear how to generalize this argument to arbitrary codomain categories and arbitrary limits/colimits as indicated by Derek's comment. (Well I already generalized to arbitrary codomain categories, I guess I mean to arbitrary shapes of limits/colimits).