Direct limit of algebras

Remarks

The relation $\sim$ is transitive and the operations do not depend on the choices of $k$ and representatives, because $I$ is directed and the $f_{ij}$ are homomorphisms. For a nullary operation $c$, take any $k \in I$ and put $c = [c^{\mathbf{A}_k}]$. The maps $a \mapsto [a]$ are homomorphisms $\mathbf{A}_i \to \varinjlim \mathbf{A}_i$, and they make the direct limit a colimit of the system.

Example: let $\mathbf{A}_n = \Z/p^n\Z$ for $n \ge 1$, with $f_{nm}(x) = p^{m-n} x$ for $n \le m$. These maps are injective homomorphisms satisfying clauses 1 and 2, and the direct limit is the Prüfer $p$-group.

Non-example: over $I = \N$, take $\mathbf{A}_n = \Z$ for all $n$ with $f_{nm}(x) = 2x$ for all $n < m$ and $f_{nn} = \id$. Clause 2 fails, since $f_{12} \circ f_{01}(x) = 4x$ while $f_{02}(x) = 2x$. Setting instead $f_{nm}(x) = 2^{m-n} x$ gives a directed system with direct limit $\Z[1/2]$.