Let \(\iota_{B}\) and \(\iota_{C}\) be the inclusion maps of \(A\) into \(B\) and \(C\), respectively. Let \(\mathcal{T}_{in}\) denote the initial topology on \(A\) induced by the set of maps \(\{ \iota_{B}, \iota_{C} \}\). By construction both vertical maps are continuous. In the left diagram, the diagonal map is continuous, hence the \(id\) map is continuous. Then in the right diagram since both vertical and horizontal are continuous, \(\iota_{B}\) is continuous. This gives \(\mathcal{T}_{B} \subseteq \mathcal{T}_{C}\).