# Terminal object in vector spaces with two fixed points (solution verification)

I am looking for solution verification for a rather basic category theory problem.

Observe a category where:

• objects are triples $$(V, v_1, v_2)$$ where $$V$$ is a vector space and $$v_1, v_2 \in V$$ are distinct vectors
• morphisms $$(U, u_1, u_2) \rightarrow (V, v_1, v_2)$$ are linear maps $$L : U \rightarrow V$$ such that $$L(u_1)=v_1$$ and $$L(u_2)=v_2$$.

The task is to find the initial and terminal objects in this category.

I worked out that $$(V, v_1, v_2)$$ is initial iff $$V$$ is two-dimensional and $$v_1, v_2$$ are linearly independent. This works since a linear map $$L: (V, v_1, v_2) \rightarrow (X, x_1, x_2)$$ given by its values on basis $$v_1, v_2$$ is unique.

I think that there is no terminal object in this category. Consider an arbitrary object $$(V, v_1, v_2)$$. If $$v_2 = \alpha v_1$$ then there are no morphisms $$(U, u_1, \beta u_1) \rightarrow (V, v_1, \alpha v_1)$$ if $$\beta \neq \alpha$$, since $$L(\beta u_1) = \beta L(u_1) = \beta v_1 \neq \alpha v_1$$ (note that $$v_1\neq 0$$ since $$v_1 \neq \alpha v_1$$ by object definition). If though $$v_1, v_2$$ are linearly independent, then again a morphism $$(U, u_1, \beta u_1) \rightarrow (V, v_1, v_2)$$ would immediately imply $$L(\beta u_1) = \beta v_1 = v_2$$, which is not the case.

I am not sure about my solution only because the problem states "to find" the object and not to prove the contrary... Also I'd be interested in other approaches, if any.