A difficult exercise
While proctoring the make-up exam for my class, I thought of an exercise related to my group theory posts on direct and free products that should cause even the more experienced mathematicians in the audience a bit of difficulty.
Consider four groups 
Use these to construct two homomorphisms from 
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Is it time for a hint?
Where to find the second homomorphism?
One comes from each universal property. One from the product and one from the coproduct.