Is this diagram of the free product of two groups correct?

Question

From Wikipedia:

"In mathematics, specifically group theory, the free product is an operation that takes two groups $G$ and $H$ and constructs a new group $G ∗ H$. The result contains both $G$ and $H$ as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from $G$ and $H$ into a group $K$ factor uniquely through a homomorphism from $G ∗ H$ to $K$."

Would this be the correct diagram showing the universal property? I'm most concerned about the red arrows. I am basing my drawing off of the kinds of illustrations I've seen of a product $X \times Y$, so do these red arrows exist and are they analogous to the projection arrows $X \xleftarrow{\pi_1} X \times Y \xrightarrow{\pi_2} Y $ that you see for products? If not, could someone do a quick drawing of the correct diagram for me? I want to put a correct diagram in my Anki cards, so any help would be greatly appreciated.

Answer 1

Red arrows should be reversed, since $G$ and $H$ are subgroups of $G*H$, so these maps are inclusions, not projections.

Although it's called free product, it's actually coproduct in category of groups, dual notion of product. Dual means that if you draw diagram for product $G\times H$, and reverse all the arrows, you'll get diagram for coproduct $G*H$.