It is easy to find examples of categories in which the coprojections of coproducts are not monic, e.g. the projection $\emptyset \times A\to A$ in $Set$ is not epic if $A$ is nonempty, so when regarded as a coprojection in $Set^{op}$ it is not monic. It is somewhat trickier to find examples of closed monoidal categories with this property, but Chu spaces give an example; see this MO question.

Last revised on June 8, 2015 at 13:50:39.
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