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Is there an inverse of the natural projection (in algebraic geometry)

I’m not quite sure how to formulate my question and have a hard time to understand the literature. But I’ll try to give some context:
I want to classify all surjective morphisms from a scheme $X$ to $\mathbb P^n$. Of course $\mathbb P^n$ comes with a natural projection $\pi \colon \mathbb P^n \to \mathbb P^1$. So, my question is whether there is an isomorphism $X \to \mathbb P^1$ making the diagram commute. In the answer it states that this indeed is the case for a scheme of finite type over a field.
But then, let me expand the question a little bit. In algebraic geometry, one defines morphisms by commutative diagrams like this:

where $Z$ is closed in $X$ and $Y$ is open in $Z$. Of course, we have a morphism of schemes $f \colon Z \to X$ and the unique morphism $g \colon X \to Z$.
So, my question is the following: If I define morphisms in a way to allow isomorphisms instead of unique morphisms, then do we get a morphism of schemes from $X$ to $Z$ when the diagram commutes? In the special case where $Z$ is the spectrum of a field and $X$ is the spectrum of a ring, the answer is yes, but I have no idea how to formalise this.

A:

Yes. The category of schemes has all equalizers. You can read about the axiom of dependent choice in Hartshorne for the development of this fact.

Fujino Art Museum

The is an art museum located in Kasukabe, Saitama, Japan. It is one of the few museums in the world that contains the paintings and works of only one artist. In the past, the museum was named “Kyuzo Fujino Art Museum” but it changed its name to reflect the fact that it houses the works of a single artist.

History
In 1918, Kyuzo Fujino began living on the
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