Wrting $\operatorname{Eq}\bigl(\prod_{i\in I}X_i\rightrightarrows\prod_{j\in J}Y_j\bigr)$ as a limit of a single diagram Let $D\colon \mathcal{I}\longrightarrow\mathcal{C}$ be a small diagram in the category $\mathcal{C}$. It is rather well known that a limit of $D$ can be computed as the equalizer of the two maps $$\prod_{i\in\mathcal{I}}D(i)\rightrightarrows\prod_{(f\colon i\to j)\in\mathcal{I}}D(j)$$
mapping a family $\{x_i\}$ to $\{x_j\}_{f\colon i\to j}$ resp. $\{D(f)(x_i)\}_{f\colon i\to j}$.
My question is wether conversely it is possible to write the equalizer of two maps of products $\operatorname{Eq}\bigl(\prod_{i\in I}X_i\rightrightarrows\prod_{j\in J}Y_j\bigr)$ as a limit of a diagram having the objects $X_i$ and $Y_j$ as its vertices.

A more elementary example.
Take the category of vector spaces over your favourite field.
Writing $V^*$ for the dual of $V$, 
$$\text{lim}_i(V_i^*)\cong(\text{colim}_iV_i)^*,$$
so every limit $\text{lim}_iV_i$ of finite dimensional vector spaces is a dual, $(\text{colim}_iV_i^*)^*$, and so can’t have countably infinite dimension.
But a countable dimensional vector space is an equalizer of maps between products of finite dimensional vector spaces.
