In a B737 NG, when a TO is flown with an Assumed Temperature N1, is the subsequent CLB N1 also adjusted (lowered) accordingly?
Possibly, but not necessarily.
The climb thrust can be selected on the FMC N1 Limit page. The choice is between CLB, CLB-1 and CLB-2. The climb thrust is then reduced according to the selection:


7 - Climb (CLB)
Push – selects full rated climb thrust limit. [...]
8 - Reduced Climb (CLB–1 and CLB–2)
Push – selects the associated reduced thrust climb mode.
CLB–1 provides a climb limit reduced by 3% N1 (approximately 10% thrust).
CLB–2 provides a climb limit reduced by 6% N1 (approximately 20% thrust).
Deletion results in the selection of CLB thrust.
Manual selection of a climb thrust rating overrides the automatic selection.

(Boeing 737 NG FCOMv2 11.40.47 - Flight Management, Navigation - FMC Preflight)
When a takeoff thrust derating or an assumed temperature is entered, the FMC automatically selects a corresponding climb derating. And yes, the climb derating is computed such that it is below or equal to the takeoff thrust:

Derated Thrust Climb
Two fixed climb thrust derates can be selected on the N1 LIMIT page. CLB–1
provides a climb limit reduced by 3% N1 (approximately 10% thrust). CLB–2
provides a climb limit reduced by 6% N1 (approximately 20% thrust). The
reduced climb setting gradually increases to full rated climb thrust by 15,000 feet.
In cruise, the thrust reference automatically changes to CRZ. The reference can be
manually selected on the N1 LIMIT page.
Use of an assumed temperature reduced thrust takeoff or takeoff derate affects the
FMCs climb derate computation. If a reduced thrust takeoff has been specified on
the TAKEOFF REF page, the FMC will re-compute CLB-1 and CLB-2 values as
required to avoid a climb N1 value greater than the reduced thrust takeoff N1
value.

(Boeing 737 NG FCOMv2 11.32.4 - Flight Management, Navigation - Flight Management Computer; emphasis mine)
However, if the pilots manually select CLB after the automatic selection of a derated climb thrust, then the climb thrust can actually increase over the TO thrust.