The `Maybe` monad is based on the endofunctor $- + 1$ (coproduct with the singleton set). Its Lawvere theory $L$ is supposed to be generated by one nullary operation (see [Plotkin and Power](http://homepages.inf.ed.ac.uk/gdp/publications/Comp_Eff_Monads.pdf)), `raise`. How do I show that the `Maybe` monad is generated from this Lawvere theory using the coend formula:
$$
T a = \int^n L(n, 1)\times a^n
$$



Lawvere theory and the Maybe monad
$L(n,1)$ consists of the $n$ projection operations $p\_i$ and the additional nullary operation $c$. So a typical element of $L(n,1)\times a^n$ has one of the two forms $(p\_i;x\_1,\dots,x\_n)$ or $(c;x\_1,\dots,x\_n)$. Th coend consists of the disjoint union of all these sets (over all $n$) modulo some identifications. When you work out those identifications, you should find that $(p\_i;x\_1,\dots,x\_n)$ gets identified with $(p\_j;y\_1,\dots,y\_n)$ when $x\_i=y\_j$, and that $(c;x\_1,\dots,x\_n)$ gets identified with $(c;y\_1,\dots,y\_n)$ regardless of the $x$'s and $y$'s. So elements of the coend arising from $(p\_i;x\_1,\dots,x\_n)$ can be regarded as just the corresponding elements $x\_i$ of $a$, while those arising from $(c;x\_1,\dots,x\_n)$ can be regarded as one extra element. In other words, you get $a+1$.


