**Motivation**: In his utterly famous paper, Rezk (here, (pag. 7)) defines a structure called "Quillen ring". I'm wearing my algebraist's hat today, so I was wondering if this definition is chosen to suggest that somewhere a "true" ring structure is hidden: now I'm in particular interested in dualizing the definition to obtain the "pushout-product" arrow, which I would like to *taxonomize* here with your kind help. [What I'm kindly asking here is: does the "pullback-exponential" operation give a "ring operation", in some reasonable sense? Does its dual give a co-ring operation?]

Let me start from the beginning. For the moment I don't even need a model structure, as I am not interested in cofibration-preservation properties (which can be used to define in a compact way the notion of "left Quillen bifunctor", as I learned about an hour ago).

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Take your favourite finitely bicomplete category $\cal C$ (in particular I'll ask for pullbacks and pushouts).

Define a binary operation on the set (of isomorphism classes) of arrows in $\cal C$, taking $f\colon A\to B$ and $g\colon C\to D$ and sending them to the arrow $$\newcommand{\diam}[4]{\left(#1 \times #4\right)\coprod_{#1 \times #3}\left( #2 \times #3 \right)} f\diamond g\colon \quad \diam{A}{B}{C}{D}\longrightarrow B\times D$$ I would like to unravel the algebraic properties of this operation $\diamond \colon {\rm Mor}({\cal C})\times {\rm Mor}({\cal C})\to {\rm Mor}({\cal C})$.

- Is it associative?
- Is it commutative?
- It seems to have a neutral element (the arrow $\varnothing\to 1$).
- Is there an operation on which $\diamond$ distributes over?

Associativity seems painful, as it is linked to simplification properties which I'm not able to deduce from easy arguments.

This is the scary form of the isomorphism which proves the associativity $f(gh) \cong (fg)h$ for $f\colon A\to B,g\colon C\to D, h\colon V\to W$ $$ \diam{\diam{A}{B}{C}{D}}{B\times D}{V}{W} \quad\cong \quad \diam{A}{B}{\diam{C}{D}{V}{W}}{D\times W} $$ Great honour to whom will be able to simplify it!