Can I glue the X axis to the Y axis?

Consider the following diagram of algebraic varieties:

$$mathbb{A}^0 to mathbb{A}^1 rightrightarrows mathbb{A}^2$$

Here the first arrow is the inclusion of the origin into the line, and the next two are the inclusion of the line into the plane as the X and the Y axes.

Does this diagram have a colimit in the category of schemes?

(The first arrowing giving inclusion of the point is not relevant; it just so happens that it was there when I met this question.)