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You might want to take a look at Problem 1 on the following problem set from a course on homological algebra.
http://www.math.stonybrook.edu/~jstarr/M536f15/M536f15ps10.pdf
In particular, the mapping complex satisfies a property (up to homotopy) that makes it seem like a kernel, and it simultaneously satisfies a property (up to homotopy) that makes it seem like a cokernel.