Let $M, N$ be right and left $A$-modules respectively. Then we have actions $$ \varphi_M: M \times A \to M, \\ \varphi_N: A \times N \to N. $$
There is a definition of $M \otimes_A N$ as follows. We have $\varphi_M \times \varphi_N: M \times A \times A \times N \to M \times N$ and $1 \times {\bf m} \times 1: M \times A \times A \times N \to M \times A \times N$. Here ${\bf m}: A \times A \to A$ is the multiplication map. $M \otimes_A N$ is defined to be the pushout of $\varphi_M \times \varphi_N$ and $1 \times {\bf m} \times 1$. Is this definition of $M \otimes_A N$ equivalent to the usual definition of $M \otimes_A N$? Thank you very much.