In my course on von Neumann algebras we have introduced the tensor product and its universal property. We have notated the morphism from the product into the tensor product with a $\hookrightarrow$. I am used to use this symbol for monomorphisms but this is clearly not one.
I did some research and figuered out that this morphism together with its target object is somehow special and is called a universal morphism. Another example for this type of morphism-object-pair can be found in the homomorphism theorem for groups, where we have that for a normal subgroup $N$ the projection onto the quotient $G/N$ is universal for the morphisms $f$ with $\ker f \subseteq N$. (universal here means that the other morphisms factor uniquely through the universal one)
It seems natural to me that one wants to emphasize in a diagram that a morphism is universal. As we did it with the $\hookrightarrow$ seems not as the best option to me. I saw a diagram for the homomorphism theorem on wikipedia but they used a normal arrow symbol. Now my question is: Are there some conventions on the notation of universal morphisms?
Kind regards, Sebastian
If have not seen any specialized notation for this before. The convention is to use a plain arrow as far as I know.
See "Categories for the Working Mathematician" (MacLane) III.1 or "Handbook of Categorical Algebra I" (Borceux) 3.1 where adjunctions are explained; or virtually any other book dealing with adjunctions (defining the notion of a universal morphism is not so common I guess).
Although in the case of an adjunction the universal morphisms are components of the unit, which is commonly denoted by $\eta$.