# A monoidal category that preserves subobjects

Let $$X$$, $$Y$$ be objects in a monoidal category $$\mathcal{C}$$, s.t. the functors $$X \otimes \_$$ and $$\_\otimes Y$$ preserve monomorphisms. Moreover, let $$A \hookrightarrow X$$, $$B \hookrightarrow Y$$ be subobjects and $$P$$ be a pullback of the cospan $$X \otimes B \hookrightarrow X \otimes Y \hookleftarrow A \otimes Y.$$ Do we have $$P \cong A \otimes B$$? If not, what do we have to assume on $$\mathcal{C}$$? On the functors $$X \otimes \_$$ and $$\_\otimes Y$$? It clearly follows when $$\_ \otimes \_$$ preserves pullbacks of monos, but this seems a bit too much.

• It works for Set as a Cartesian monodial Category. – user458276 Feb 15 at 14:00