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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.

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

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