If I have a monad $$ U \stackrel{\alpha}{\longrightarrow} V \stackrel{\beta}{\longrightarrow}W $$ then there should be a natural map $$ \text{cokernel}(\alpha) \rightarrow W $$ but I can't think of what it is. This might be dependent on looking at the display to see more maps, but I'm not sure. It might not actually have anything to do with monads either, that's just the context I'm looking at. Thanks for the help.
2025-01-13 07:45:03.1736754303
Natural map from cokernel of a monad
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A monad is in particular a complex, meaning that $\beta \circ \alpha = 0$. Thus, $\operatorname{im} \alpha \subseteq \ker \beta$. Use the fundamental theorem on homomorphisms.