(Notations are related to Riemannian geometry and PDE theory, and have their usual meanings.)
Theorem 4.2 (scalar maximum principle, second version: lower bounds are preserved for supersolution): Let $g(t): t\in [0, T)$ be a $1$-parameter family of Riemannian metrics and $X(t) : t\in [0,T)$ a $1$-parameter family of vector fields on a closed manifold $M^n$. Let $u : M \times [0, T)\to\mathbb R$ be a $C^2$ function. Suppose that there exists $\alpha \in\mathbb R$ such that $u (x, 0)\ge \alpha$ for all $x\in M$ and $\color{red}{\text{that}}$ $u$ is a supersolution of the heat equation $\color{red}{\text{at any}}$ $(x, t) \in M \times [0, T)$ such that $\color{red}{u (x, t) < \alpha}$. Then $u (x, t) \ge \alpha$ for all $x \in M^n$ and $t \in [0,T)$.
Note: $u$ is a super solution of the following heat equation \begin{equation}\label{eq1} \frac{\partial v}{\partial t} = \Delta_{g(t)}v+\langle X,\nabla v\rangle. \end{equation} This means $u$ satisfies \begin{equation}\label{eq2} \frac{\partial u}{\partial t}(x,t) \ge (\Delta_{g(t)}u)(x,t)+\langle X,\nabla u\rangle(x,t). \end{equation}
This version of maximum principle can be found in the book The Ricci flow: An Introdution by Chow and Knof, Chapter - $4$.
I am confused with the statement of the above theorem. My question is: First we assume that there is a constant $\alpha\in\mathbb R$ for which the function $u(x,t)$ is bounded below by $\alpha$ at $t=0$ (in notation $u(x,0)\ge \alpha$) for all $x\in M$. With an "and" we make the second assumption: the same function $u$ is a supersolution of the above mentioned heat equation at any $(x,t)\in M\times [0,T)$ such that $u(x,t)<\alpha$. What does that assumption mean? If at any point $(x,t)$ the function $u$ is less than $\alpha$ then (the conclusion) how $u(x,t)$ is greater equals to $\alpha$ for all $(x,t)\in M\times[0,T)$? I am sure that I am missing something but can not figure out what am I missing. Kindly help me understand this.
I have seen the posts Proof of weak maximum principle for heat-type equations, Argument in a proof for scalar maximum principle but I didn't get enough help. Thanks in advance.