Lemma 40.1:

Suppose that $\{M_{\alpha}\;/\; \alpha \in \Lambda\}$ is directed system in one of the categories Gr, AbGr or Top, and for some pair $x_{\alpha}\in M_{\alpha}$ and $x_{\beta}\in M_{\beta}$ there exists $\gamma \in \Lambda$ such that $f_{\alpha\gamma}(x_{\alpha}) = f_{\beta\gamma}(x_{\beta})$, then for every $\delta \geq \gamma$, $f_{\alpha\delta}(x_{\alpha}) = f_{\beta\delta}(x_{\beta})$.