Why colimits do not commute with tensor products?

Question

A more intuitive question: In may minmal knowledge about category, I know that any two colimits of a category commute. In the category of $A$-modules over a commutative ring $A$ with unit, the tensor product is such a colimit. But so are the image and the cokernel. But in general, we do not have $\mathrm{im}(f \otimes_A g) \cong \mathrm{im}(f) \otimes_A \mathrm{im}(g)$ or $\mathrm{coker}(f \otimes_A g) \cong \mathrm{coker}(f) \otimes_A \mathrm{coker}(g)$ for two $A$-linear maps $f$ and $g$ (e.g. here or here). What do I miss?

Answer 1

If we have two short exact sequences $$0\to A_i\xrightarrow{\iota_i}B_i\xrightarrow{\pi_i}C_i\to0,$$ $i=1, 2$, we may show that $\pi_1\otimes\pi_2\colon B_1\otimes B_2\to C_1\otimes C_2$ is surjective, and thus fits in a SES $$0\to K\xrightarrow{k}B_1\otimes B_2\xrightarrow{\pi_1\otimes\pi_2}C_1\otimes C_2\to0.$$ Consequently, the property "is a cokernel" is preserved, but the "is a cokernel of $\iota_1\otimes\iota_2$" is not guaranteed. In other words, "left adjoints preserve colimits" still works. Hope this helps :)