category of ideal pairs of rings

The category of ideal pairs of rings is defined as category containing

1. pairs \((R, \mathfrak{a})\) where \(R\) is a ring, \(\mathfrak{a}\) an ideal
   
2. morphisms consist of ring-homomorphism \(\varphi: R \rightarrow S\) such that \(\varphi[\mathfrak{a}] \subseteq \mathfrak{b}\)