I'm trying to understand the idea of an Albanese variety. It reminds me of something simpler:

Given a set $X$ with a chosen point $x \in X$, we can form the free abelian group on the pointed set $(X,x)$, which is just the free abelian group on $X$ modulo a relation saying $x = 1$.

If we call this group $A(X,x)$, it has a nice universal property: any map of pointed sets $f$ from $X$ into an abelian group $A$ factors uniquely as the obvious inclusion

$$i_X \colon X \to A(X,x)$$

followed by some homomorphism

$$\overline{f} \colon A(X,x) \to A.$$

So:

$$ f = \overline{f} \circ i_X $$

The process of taking the free abelian group on a pointed set defines a functor

$$ A \colon \mathrm{Set}_* \to \mathrm{AbGp} $$

which has a right adjoint

$$ U\colon \mathrm{AbGp} \to \mathrm{Set}_* $$

sending any abelian group $A$ to its underlying pointed set $(A,1)$. The composite

$$ U A \colon \mathrm{Set}_* \to \mathrm{Set}_* $$

is thus a monad, and if I'm not mistaken, the algebras of this monad are just abelian groups.

The idea of an Albanese variety seems to be similar, but working with projective algebraic varieties instead of sets. Namely:

Given any such variety $X$ with a chosen point $x$ there is an abelian variety called the Albanese variety $A(X,x)$, apparently defined by the following universal property: there is a map of varieties

$$i_X \colon X \to A(X,x)$$

such that any map of pointed varieties $f$ from $X$ into an abelian variety $A$ factors uniquely as $i_X$ followed by some map of abelian varieties

$$\overline{f} \colon A(X,x) \to A.$$

So:

$$ f = \overline{f} \circ i_X $$

(The Wikipedia article on Albanese varieties doesn't clearly require that $\overline{f}$ be a map of abelian varieties, but Ravi Vakil's lecture notes do, so I'm going with that.)

So, naturally, I'm wondering: does taking the Albanese variety define a functor from pointed varieties to abelian varieties, which has a right adjoint, which together define a monad on the category of pointed varieties whose algebras are the abelian varieties?

The so-called Albanese map $i_X \colon X \to A(X,x) $ would then be the unit of this monad, while the multiplication of the monad would be the map of abelian varieties $\overline{1} \colon A(A(X,x),x) \to A(X,x) $ obtained from the identity map of varieties $1 \colon A(X,x) \to A(X,x)$.