Step 1: Build a truth table instead of tracing the circuit case by case in words.
Take HIGH = logic 1 and LOW = logic 0 for inputs $A$ and $B$, and note the diodes only conduct when forward biased, pulling the shared output node down toward whichever input is LOW.
Step 2: Fill in all four input combinations.
For $A,B = 0,0$ the output is pulled LOW (1). For $0,1$ and $1,0$, one diode still conducts and pulls the output LOW (1). Only for $A,B = 1,1$ do both diodes stay reverse biased, letting the output float HIGH through the pull-up, giving output 0.
Step 3: Read off the pattern.
The output sequence $1,1,1,0$ for inputs $00,01,10,11$ is exactly the AND output inverted. \[ \boxed{\text{NAND gate}} \]