The Stacks project

Exercise 111.2.2. Let $I$ be a directed set and let $(A_ i, \varphi _{ij})$ be a system of rings over $I$. Show that there exists a ring $A$ and maps $\varphi _ i : A_ i \to A$ such that $\varphi _ j \circ \varphi _{ij} = \varphi _ i$ for all $i \leq j$ with the following universal property: Given any ring $B$ and maps $\psi _ i : A_ i \to B$ such that $\psi _ j \circ \varphi _{ij} = \psi _ i$ for all $i \leq j$, then there exists a unique ring map $\psi : A \to B$ such that $\psi _ i = \psi \circ \varphi _ i$.