Let $\mathfrak C$ be a category with products. Given a family $(X_i)_{i \in I}$ of objects of $\mathfrak C$, we get their product $\prod_{i \in I} X_i$ with projections $\pi_i : \prod_{i \in I} X_i \to X_i$.
Now let $\mathcal F(I)$ denote the set of all finite subsets of $I$ which is partially ordered via $J \le K$ iff $J \subset K$. The partially ordered set $\mathcal F(I)$ is directed.
For each $J \in \mathcal F(I)$ we can form the subproduct $\prod_{i \in J} X_i$ of $\prod_{i \in I} X_i$. If $J, K \in \mathcal F(I)$ and $J \le K$, we get a canonical projection morphism $\pi_{K,J} : \prod_{i \in K} X_i \to \prod_{i \in J} X_i$. The collection all these objects and morphisms $$(\prod_{i \in J} X_i, \pi_{K,J})$$ is easily seen to be an inverse system over $\mathcal F(I)$. It seems evident that this inverse system has an inverse limit with limit object $\prod_{i \in I} X_i$.
Is there a reference for this fact? What is the proof?
Dually we can consider a category with coproducts (sums). Given a family $(X_i)_{i \in I}$ of objects, we get their coproduct $\coprod_{i \in I} X_i$ with "embeddings" $\iota_i : X_i \to \coprod_{i \in I} X_i$. In analogy to the above case we get a collection $$(\coprod_{i \in J} X_i, \iota_{J,K})$$ which is easily seen to be a direct system over $\mathcal F(I)$. Then a "dual" proof will show that this direct system has a direct limit with limit object $\coprod_{i \in I} X_i$.
Here is a proof for the product case; the coproduct case follows by considering the dual category.
For each $M \subset I$ let $\Pi(M) = \prod_{i \in M} X_i$ with projections $\pi_{M,i} : \Pi(M) \to X_i$ for $i \in M$. Note that $\Pi(I) = \prod_{i \in I} X_i$ is the "full" product and the $\Pi(M)$ are subproducts of $\Pi(I)$. The projections $\pi_{I,i} : \Pi(I) \to X_i$ are usually denoted as $\pi_i$. For $M = \{i\}$ we agree to take $\Pi(\{i\}) = X_i$ with $\pi_{\{i\},i} = id_{X_i}$.
If $M \subset N$, the projections $\pi_{N,i} \to X_i$ with $i \in M$ induce a unique morphism $\pi_{N,M} : \Pi(N) \to \Pi(M)$ such that $\pi_{M,i} \circ \pi_{N,M} = \pi_{N,i}$ for all $i \in M$. This morphism is the canonical projection onto the subproduct. Due to our above convention we have $\pi_{M,\{i\}} = \pi_{M,i}$. For $N = I$ we simply write $\pi_J = \pi_{I,J}$.
If $M \subset N \subset P$, we have $\pi_{N,M} \circ \pi_{P,N} = \pi_{P,M}$:
In fact, for $i \in M$ we have $\pi_{M,i} \circ (\pi_{N,M} \circ \pi_{P,N}) = (\pi_{M,i} \circ \pi_{N,M}) \circ \pi_{P,N} = \pi_{N,i} \circ \pi_{P,N} = \pi_{P,i} = \pi_{M,i} \circ \pi_{P,M}$.
This gives an inverse system $$\mathfrak I = (\Pi(J), \pi_{K,J} : \Pi(K) \to \Pi(J))$$ over $\mathcal F(I)$. We claim that $\varprojlim \mathfrak I $ exists and is given by $$\varprojlim \mathfrak I = (\Pi(I),\pi_J) .$$ To prove it, first observe that $$\pi_{K,J} \circ \pi_K = \pi_J \text{ for } J \le K \tag{1}.$$ We shall now verify the universal property of the inverse limit. So let $Z$ be any object of $\mathfrak C$ and $p_J : Z \to \Pi(J)$ be a family of morphisms such that
$$\pi_{K,J} \circ p_K = p_J \text{ for all } J \le K . \tag{2}$$ We have to show that there exists a unique $\phi : Z \to \Pi(I)$ such that $$\pi_{J} \circ \phi = p_J \text{ for all } J \in \mathcal F(I) .\tag{3}$$
By the universal property of the product there exists a unique $\phi : Z \to \Pi(I)$ such that $$\pi_{\{i\}} \circ \phi = \pi_{i} \circ \phi = p_{\{i\}} \text{ for all } i \in I .\tag{4}$$ It therefore suffices to show that this $\phi$ satisfies $(3)$.
The codomain of both morphisms in $(3)$ is $\Pi(J)$, thus it suffices to show that for all $i \in J$ we have $\pi_{J,i} \circ (\pi_{J} \circ \phi) = \pi_{J,i} \circ p_J$. But in fact $$\pi_{J,i} \circ (\pi_{J} \circ \phi) = (\pi_{J,i} \circ \pi_{J}) \circ \phi = \pi_{\{i\}} \circ \phi = p_{\{i\}} = \pi_{J,\{i\}} \circ p_J = \pi_{J,i} \circ p_J .$$